Update: The first issue about the "cut" is resolved. I used the wrong order for the uvadr and padr.
Now interestingly, this also seems to fix the strange looks of the Cube for some reason!
So the first and second issue are resolved by this :) You can find the corrected versions of the
functions below.

The third question remains though: How can I get the texture to be drawn over the
Cube (or above everything)? Currently, it gives a very strange effect.